2025 VJC P1 Q11

It is given that \(2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y\).

1. Use the substitution \(y = \ln \left( \frac{u}{x} \right)\) to show that the differential equation can be reduced to \(\frac{\mathrm{d}u}{\mathrm{d}x} = \mathrm{f}(u)\), where the function \(\mathrm{f}(u)\) is to be found.[3]
2. Given that \(y\) has a minimum value at \(x = 3\), solve the differential equation \(2 \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x} - \ln x = y\), to find the particular solution for \(y\) in terms of \(x\).[5]
3. Sketch the graph of this particular solution.[2]